Multiproduct static model with a limited capacity of the warehouse

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'''Russian: [http://scm.gsom.spbu.ru/index.php/Многопродуктовая_статическая_модель_с_ограниченной_вместимостью_склада Многопродуктовая статическая модель с ограниченной вместимостью склада]'''
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Multiproduct static model with a limited capacity of the warehouse
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Consider the case of static demand.
Consider the case of static demand.

Revision as of 11:59, 20 August 2011

Russian: Многопродуктовая статическая модель с ограниченной вместимостью склада


Multiproduct static model with a limited capacity of the warehouse

Consider the case of static demand.

The basic assumptions of the model

  • consider the problem of managing multiple types of stock;
  • storage space is limited;
  • order cost is a constant;
  • price of the resource unit is constant;
  • carrying cost of resource unit is a constant;
  • затраты на оформление, связанные с размещением заказа, - постоянная величина (константа);
  • order cost is a constant;
  • no deficit.

The basic notations

There are two types of inventory \, i, \, i=1,2,...,n:

  • \, D_{i} - demand rate;
  • \, h_{i} - marginal carrying costs;
  • \, K_{i} - fixed order costs;
  • \, TCU(y) - total costs per unit of time;
  • \, y_{i} - order quantity;
  • \, y_{i}^{*} - economic order quantity;
  • \, a_{i} - the space required to store the unit;
  • \, A - the maximum space required to store \, n types of resources.

Inventory optimal control

According to the assumptions of the model, consider dynamics of the resource stock \, i, fig. 1:

Z3pic1.JPG

Fig. 1.' Dynamics of changes in resource stock \, i.


Multiproduct static model with a limited capacity of the warehouse can be formalized as a nonlinear programming problem:

\min TCU(y_{1} ,y_{2} ,...,y_{n} )=\sum \limits _{i=1}^{n}\left(\frac{K_{i} D_{i} }{y_{i} } +\frac{h_{i} y_{i} }{2} \right)

\sum \limits _{i=1}^{n}a_{i} y_{i}  \le A

\, y_{i} >0

\,i=1,2,...,n.

Inventory optimal control

For the above problem of nonlinear programming Lagrange function has the form:

L(\lambda ,y_{1} ,y_{2} ,...,y_{n} )=TCU(y_{1} ,y_{2} ,...,y_{n} )-\lambda \left(\sum \limits _{i=1}^{n}a_{i} y_{i}  -A\right)=

 =\sum \limits _{i=1}^{n}\left(\frac{K_{i} D_{i} }{y_{i} } +\frac{h_{i} y_{i} }{2} \right) -\lambda \left(\sum \limits _{i=1}^{n}a_{i} y_{i}  -A\right) ,

where \, \lambda <0 is a Lagrange multiplier.

The Lagrange function for multiproduct static model with a limited capacity of the warehouse is convex, hence, the optimal value \, \lambda and \, y_ {i} can be found from the first order conditions:

 \frac{\partial L}{\partial \lambda } =-\sum \limits _{i=1}^{n}a_{i} y_{i}  +A=0 (limitation on the capacity of a warehouse at the optimal point);

\frac{\partial L}{\partial y_{i} } =-\frac{K_{i} D_{i} }{y_{i}^{2} } +\frac{h_{i} }{2} -\lambda a_{i} =0 .

The solution of the second equation is:

y_{i}^{*} =\sqrt{\frac{2K_{i} D_{i} }{h_{i} -2\lambda ^{*} a_{i} } } .

The optimal solution value \, \lambda^{*} with the desired accuracy can be found as follows:

1. Set the initial value \, \lambda =0

2. Set the value \, \varepsilon for decreasing the value \, \lambda (accuracy)

3. Consistently reduce \, \lambda on the value of  \, \varepsilon , substituting the value of \, \lambda in  y_ {i} = \sqrt {\frac {2K_ {i} D_ {i}} {h_ {i} -2 \lambda a_ {i}}} and checking the performance limitations on the capacity of the warehouse.


The optimal strategy for inventory management in our model has the form:

Step 1. Calculate the optimal volume of orders, not including the restriction on the capacity of storage (see Basic economic order quantity) as follows:

y_{i}^{**} =\sqrt{\frac{2K_{i} D_{i} }{h_{i} } },

\, i=1,2,...,n.

Step 2. Subject to the values found \, y_ {i }^{**}, \, i = 1,2 ,..., n verify constraints on the capacity of the warehouse. If this restriction is satisfied, then the set of values \, y_ {i }^{*}, \, i = 1,2 ,..., n is the optimal solution for multiproduct static model with a limited capacity of the warehouse. Otherwise, the best solution is the set \, y_ {i }^{*}, \, i = 1,2 ,..., n

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